According to Socrates interpreted by Plato (in the Republic), the government by the best. This is a fine idea, except that who is to know the best? Plato wanted government to be in the hands of those trained to to rule by a special education. Perhaps partially achieved, or at least attempted, in the English Public (Private) Schools, the graduates of which ran the British Empire. This example may disprove Plato's theory or show that it is not attainable.
In practice those called aristocrats in most European countries usually turn out to be the richer, the descendants of the German tribesmen who invaded the Roman Empire and seized the land, ruling over the "natives" - Celts and Romans - but not obviously better rulers. In Europe the descendants of the German invaders became the feudal lords. These in turn became the later aristocracy. The result was oligarchy.
In Britain the aristocracy claimed to be descendants of the Normans, a mixed group, some of whom could trace their ancestry to Vikings, while others descended from the Britons who left Britain when the Saxons arrived, settled in Brittany but returned with William the Conqueror.
Socrates may have implied the common idea that some people achieve greater understanding than the average, perhaps by following a special course of study and learning experiences. However, it may be a condition of achieving this knowledge that they do not claim powers publicly. Thus, paradoxically, a true Socratean aristocracy would not rule in an obvious way and would not be called Aristocracy. In practice Aristocracy descends in families; knowledge descends from teacher to student.
In Muslim countries the descendants of the Prophet, the Hashemites, may be considered a kind of aristocracy. Some of these are actual rulers: Jordan, Morocco, Iran (in the past). In apparently democratic states, such as the United States and India certain families provide the rulership. In the US the Bushes and the Kennedys seem to be at least dynasties. In India the leading family are the descendants of the first Prime Minister, Nehru. Do they constitute an informal aristocracy? None of these families seem especially fit to rule.
In Rwanda and Burundi the Tutsis are sometimes thought to have been an aristocracy. They were once considered to be the descendants of a nomadic group, probably of Nilotic speakers who ruled over a Bantu-speaking peasantry. (This theory has been discredited in recent years by the absence of evidence to support it. See Interlacustrine)
The European system of aristocracy was fairly closed - except in Britain, where ennoblement of successful people from the professions and business has been going on for centuries. A few families may claim to be descended from the Normans who seized the land in 1066, but most are in fact English, or indeed just ordinary families who acquired wealth and rose in importance, generation by generation.
The French aristocracy, so many of whom were killed during the early years of the Revolution, had not played a role in the state since the early 17th century. After the failure of an aristocratic revolt against the king - the Fronde - they were excluded from power. Instead their lives were confined to spending money on pleasure. It is not a surprise that their profligacy was bitterly resented in 1789 when the state was bankrupt. Can one compare them with the Wall Street bankers of 2008 whose life style was similar while, like the French aristos, not obviously contributing to the public good?
Until the first world war the Aristocracy had a grip on the government of Britain and its empire as an informal oligarchy; after that war they lost control as the governors came from wider areas of society. One reason of course was that many died in the war, and the vote had been extended to every adult. Until the 20th century the House of Lords seemed to be more important than the Commons. That came to an end with Lloyd George's budget of 1909 when the Lords voted against it, breaking the convention that the Lords never opposed a finance bill. After a campaign of resistance they had to acknowledge defeat when, after two general elections to the Commons in 1910 each with a Liberal victory, they had accept that the Liberal government had the permission of the king to create enough new Peers to vote the bill through. Since then the Commons has been supreme.
Death duties were introduced to prevent the wealthy handing on their wealth to their sons. At that time the Parliament Act ruled that the Lords could only delay legislation, not veto it.
Now that most of the hereditary members of the House of Lords have been excluded, a formal aristocratic role in British government has apparently ended. (Nevertheless, the current (2011) "coalition" Cabinet in Britain is composed mainly of people from Eton and other Public Schools, and generally wealthy.)
Note on terminology
Plato - The Republic
Patrick Leigh Fermor - Between the Woods and the Water
a walking tour of central Europe shortly before the second world war, a classic. Among the people he meets are aristocrats in several countries, including Hungary.
Between the Woods and the Water: On Foot to Constantinople: From The Middle Danube to the Iron Gates (New York Review Books Classics)
Zwischen Wäldern und Wasser: Zu Fuss nach Konstantinopel: Von der mittleren Donau bis zum Eisernen Tor. Der Reise zweiter Teil
Count Heinrich Coudenhove-Kalergi - Antisemitism throughout the ages (1901)
An Austrian aristocrat, related to the Greek, Russian, Dutch, British and Japanese aristocracy, argues against the prevailing antisemitism of his class, reprinted in 1935 as an answer to Hitler.
No longer in print in English but worth finding
in a library.
The Kalergis are examples of the inter-connectedness of the European aristocracy.
Ursula le Guin - Orsinian Tales
Ursula le Guin - Orsinian Tales
Orsinian Tales: Stories
David Cannadine - the Decline and Fall of the British aristocracy